New constants in pretabular superintuitionistic logics: P. Novikov's approach

In the late fifties of the twentieth century a problem was posed by P. S. Novikov concerning new logical connectives as extranotions for a language with standard logical connectives $\vee$, $\wedge$, $\rightarrow$, $\neg$. Ya. S. Smetanich has given exact formulations for approach of Novikov to the concept of new logical connectives in superintuitionistic logics (new logical connective, Novikov completeness). In the present paper, the Novikov problem concerning new additional constants is considered in pretabular superintuitionistic logics $LC,$ $L2,$ $L3$: the logic of chains, the logic of rooted frames of the depth not exceeding $2$ (fans), the logic of rooted frames with the top and with the depth not exceeding $3$ (diamonds). The classification for the family of all Novikov-complete extensions of the pretabular superintuitionistic logics in a language containing additional logical constants is described. For these logics, the classification is obtained in the terms of finite frame with coloring: in the language with several additional constants for $LC$ and $L2$ and with a single additional constant for $L3$. Decidability of the (algorithmic) conservativeness problem for extensions of all pretabular superintuitionistic logics is established. The algorithmic problem of conservativeness recognition is investigated.